Calculate P(A or B) for mutually exclusive, non-exclusive, and three-event scenarios with stacked probability bars, comparison tables, and repeated-trial analysis.
The OR probability calculator computes P(A or B) — the probability that at least one of two (or three) events occurs. It supports three modes: mutually exclusive events (P(A∪B) = P(A) + P(B)), non-exclusive events (P(A∪B) = P(A) + P(B) − P(A∩B)), and three-event inclusion-exclusion.
Understanding OR probability is essential for risk assessment ("probability of failure A or failure B"), insurance ("probability of claim type A or B"), and everyday probability reasoning. The addition rule is one of the most commonly needed probability formulas.
This calculator visualizes the probability breakdown with stacked bars, compares all probability components side-by-side, and shows how the probability of "at least one" occurrence scales across repeated independent trials. It is a fast way to see whether you need to subtract overlap, use a complement, or switch to inclusion-exclusion for multiple events.
The addition rule is one of the most frequently needed probability calculations, from simple "what are the odds of X or Y?" questions to larger risk models with multiple failure modes. This calculator helps you avoid double-counting, compare the overlap against the union, and confirm that the parts still sum to the whole.
It is useful whenever you need a clean answer to "A or B", whether that means a coin result, a failure condition, or a compound event in a probability table.
Mutually exclusive: P(A ∪ B) = P(A) + P(B). Non-exclusive: P(A ∪ B) = P(A) + P(B) − P(A∩B). Three events (inclusion-exclusion): P(A∪B∪C) = ΣP − ΣP(pairs) + P(A∩B∩C).
Result: P(A or B) = 44%
P(A∪B) = 30% + 20% − 6% = 44%. The 6% overlap is subtracted to avoid double-counting events where both A and B occur.
For n events, the inclusion-exclusion formula alternates adding and subtracting intersections: P(A₁∪...∪Aₙ) = ΣP(Aᵢ) − ΣP(Aᵢ∩Aⱼ) + ΣP(Aᵢ∩Aⱼ∩Aₖ) − ... This formula has 2ⁿ−1 terms, making it impractical for many events. In such cases, the complement method is preferred.
System reliability often uses OR probability. A system fails if component A OR component B fails. For independent components with failure probabilities pA and pB, system failure probability = pA + pB − pA×pB. Redundant systems reverse this: the system survives if component A OR component B survives.
De Morgan's laws connect OR and AND through complements: P(A∪B) = 1 − P(A'∩B') and P(A∩B) = 1 − P(A'∪B'). These identities are fundamental for converting between "at least one" and "all/none" probability problems.
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OR (union) asks "at least one event occurs" and is P(A) + P(B) − P(A∩B). AND (intersection) asks "both events occur" and is P(A)×P(B) for independent events. OR always gives a larger or equal probability.
When they cannot happen simultaneously. A coin landing heads AND tails is impossible. A single card being both a heart and a spade is impossible. For mutually exclusive events, P(A∩B) = 0.
It divides the total probability space into: only A, both A and B, only B, and neither. These four regions always sum to 100%.
It uses the inclusion-exclusion principle for three events. For independent events, all pairwise and triple intersections are computed from the marginal probabilities.
Each trial has probability (1−P(A∪B)) of "no occurrence." After n trials, P(none) = (1−P(A∪B))^n, which shrinks exponentially. So P(at least one) = 1 − (1−P(A∪B))^n grows rapidly.
No. P(A or B) is capped at 100%. If P(A) + P(B) > 100% and they're "mutually exclusive," the probabilities are inconsistent — genuinely mutually exclusive events can't have P(A) + P(B) > 1.